Saturday, August 27, 2011

Fractal (Information) Decay

Fractal Decay
As shown in the animation below, and described in my earlier entries, the fractal demonstrates development and growth: but if this reversed, it also demonstrates decay. Developing from the  first simple iteration to the complex, and in reverse, decay from the complex to the simple; from the snowflake to the triangle.

fractal growth and development

Below are two diagrams that analyse the Koch Snowflake fractal: the top diagram shows the exponential, and the lower the log. Both are split vertically (with a 'black' line of reflection) - showing development on the left side and decay on the right side.  - keep your eye on the snowflake.. 

The blue curve shows the extra benefit of another iteration (in terms of Area), and green - the extra cost of  producing or iterating. 
As fractal development is exponential, it follows that so is decay. The above diagram of the two shows the exponential curve - with a constant elasticity (or sensitivity),  and the below the logarithmic line of the above - exposing the (economic) elasticities.
Benefit Decay Curve
Rising blue benefit decay curve:
This curve is the opposite to the downward sloping benefit curve and demonstrates that benefit increases as the fractal (snowflake) decays. It is the marginal or change in the Total Area of the snowflake (Total Area is not shown here, see my early posts to see this). It is not to say that the Total benefit of loss gets greater, but that the change gets greater - the change (or loss) is small in the beginning and greater later.
Decreasing green decay cost curve
The cost, the pain, is highest at the beginning (assuming beginning is at equilibrium) and there after falls away.

Decay Elasticity

Elasticity of Blue decay curve:
Inelastic at (or near) the centre (the equilibrium point) suggests another iteration (or one less iteration in the case of decay) will not change the shape or Area greatly. Conversely, elastic at the end of decay suggests noticeable, or sensitivity to change in Area.

Elasticity of green decay curve
Inelastic suggests insensitive to change in the beginning - the pain, the cost remains, it lingers, the shape remains, the memory remains.
Elastic suggests after time the converse to the above. The shape is a distant memory and is changing - no pain.

Where do we see or experience this decay in reality?
  • Think of a dying tree - the branches wasting away to expose only the 'trunk'.
  • What do we remember after time - just the basics,
  • the pain is in the beginning.
  • The melting of a snowflake.
  • It makes me think of Lord Rutherford's radioactive decay, carbon dating and transmutation??
  • Viewing something from a distance.
  • Is this what is termed entropy?